Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

124 kenneth manderscase is occasionally described in part: ‘as in the diagram’. Both the diagramand the discursive text appear to have the resources to record such co-exactcontents.It is in fact not immediately clear what decisive virtue favors fully discursiveformulation. Proclus’ commentary is a teaching text; and it is no doubt usefulto sensitize the student to the relationship beween diagram and text. Butbeyond the pedagogical role of attending to such refinements, one can still askwhether continued pursuit of explicit verbal statement beyond the level attainedin Euclid’s time is a genuine contribution to the effectiveness of geometricalpractice; or just the product of a scholastic philosophical commentary tradition,out of touch with what matters in geometry, and in single-minded pursuit ofdiscursively complete argument.We shall return to explain why a diagram does not function well as partof the statement of a proposition. But on the surface, leaving unambiguouslyreadable contiguity specifications implicit in the initial diagram of a propositionwould seem not to pose undue risk of disarray. For an initial diagram can bedesigned to show a ‘clear case’, avoiding diagram sensitivities that might causean appearance control problem to limit our ability to read off its contiguitiesin the diagram. On the other hand, traditional practice is plainly limited in itsartifact support of what we have called appearance control (limited by our lackof metric control of diagrams individually) and case branching control (limitedinstead by our lack of intellectual grip on the totality of diagrams satisfyingconditions laid down in the text, quite independently of limitations of metriccontrol). Because traditional diagram-based geometrical reasoning is somewhatopen-ended in both these ways, it requires a sustained critical exploration ofdiagram use.We probe case branching control in an argument by feeling around fordiagram variants which satisfy the conditions laid out in the text. Attemptinga more principled distinction between case and objection than that givenby Proclus, we might propose to call a variant recognized as arising fromprobing case branching control a case; and one recognized as arising fromprobing appearance control, an objection. The variants, for example, in theall-triangles-are-isoceles argument arise from probing appearance control, notcase branching control. For they purport to depart from the same initial metricchoices by fully determinate construction; any inability on our part to tellwhich is appropriate is a breakdown of appearance control.By this standard, Dubnov mischaracterizes his response to the challenge:what is called for is not that we ‘consider all cases’ (p. 24) but that we, givenone of the diagrams, probe so as to come up with an account of its appropriateappearance. This might, however, take the form of surveying the variants viable